Synchronization and small-signal analysis of nonlinear periodic circuits

Common approaches to simulate the steady-state behavior of nonlinear periodic circuits forced by a periodic signal of small amplitude assume that the forcing signal effects are additive to the steady-state solution of the unperturbed circuit. This assumption leads to the adoption,of the variational model of the nonlinear unperturbed circuit. The variational model does not pose any particular problem when dealing with nonautonomous circuits, but must be suitably formulated when autonomous circuits are considered and the frequency of the forcing signal is close to the working frequency of the unperturbed nonlinear circuit. We show that, in this case, synchronization effects must be accounted for, and, as synchronization phenomena are intrinsically non- linear, it is impossible to take them into account using a variational model. In fact, conventional variational models are derived from the unperturbed nonlinear circuit working at steady state and with a fixed relative phase between perturbation and system, i.e., without any possibility of phase shifts (that is, of any dynamics leading to possible synchronization). In general, this yields inaccurate or even wrong results. In this paper, we investigate this limitation of the approaches based on the variational model. Some simulation results are reported that show the transition between the nonsynchronization region to the synchronization one of well-known simple oscillators, such as the Van der Pol one when the parameters of the small-signal perturbation are varied.